Hawking from Catalan

Abstract

The Virasoro algebra determines all `graviton' matrix elements in AdS${}_3$/CFT${}_2$. We study the explicit exchange of any number of Virasoro gravitons between heavy and light CFT${}_2$ operators at large central charge. These graviton exchanges can be written in terms of new on-shell tree diagrams, organized in a perturbative expansion in $h_H/c$, the heavy operator dimension divided by the central charge. The Virasoro vacuum conformal block, which is the sum of all the tree diagrams, obeys a differential recursion relation generalizing that of the Catalan numbers. We use this recursion relation to sum the on-shell diagrams to all orders, computing the Virasoro vacuum block. Extrapolating to large $h_H/c$ determines the Hawking temperature of a BTZ black hole in dual AdS${}_3$ theories.

19+8 pages, 5 figures

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• Towards a Bootstrap approach to higher orders of epsilon expansion
• New Properties of Large-$c$ Conformal Blocks from Recursion Relation
• One-Loop Transition Amplitudes in the D1D5 CFT
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